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Homework Help: Find values for which the limit exists

  1. Oct 16, 2005 #1
    Hi,
    I'm given this problem:

    Find conditions for variables a, b, c so that the limit

    [tex]
    \lim_{[x,y] \rightarrow [0,0]} \frac{xy}{ax^2 + bxy + cy^2}
    [/tex]

    exists.

    What I have only found so far is that for all variables non-zero the limit doesn't exist. Anyway, I have no clue how to find the conditions for which it does. I tried a = b = c = 0, but it doesn't seem to help to me...

    Thank you for the enlightenment.
     
  2. jcsd
  3. Oct 16, 2005 #2

    Lisa!

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    Gold Member

    OOps I was too late in deleting my post. Actually I made a mistake in solving that and yes, what makes me unsure is that I don't think we could use hopital rule for these kind of limit.
     
  4. Oct 16, 2005 #3

    Lisa!

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    Gold Member

    You'd better to ask another homework helper, but I solve it in another way now . If you look at numerator nad denominator, you see both of them have xy. So?
     
  5. Oct 16, 2005 #4
    Ok, now it seems to me that the condition for the limit to exist is that a = c = 0.

    Anyway, it is just the result of guessing method, is there any more exact approach to solve this?
     
  6. Oct 16, 2005 #5

    HallsofIvy

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    Science Advisor

    For functions like this, where you have two variables, I find it best to convert to polar coordinates. That way, exactly one variable, r, measures the distance to (0,0) which is the crucial factor. In polar coordinates,
    [itex]x= r cos(\theta)[/itex] and [itex]y= r sin(\theta)[/itex] so
    [itex]xy= r^2 cos(\theta)sin(\theta)[/itex], [tex]x2= r^2 cos^2(\theta)[/itex], and [tex]y^2= r^2 sin^2(\theta)[/itex].
    Of course, then [itex]ax^2+ bxy+ cy^2= ar^2cos^2(\theta)+ br^2sin(
    theta)cos(\theta)+ cr^2sin^2(\theta)[/itex] so that
    [itex]ax^2+ bxy+ cy^2= r^2(acos^2(\theta)+ bsin(
    theta)cos(\theta)+ csin^2(\theta)[/itex].

    That means that
    [tex]\frac{xy}{ax^2+ bxy+ cy^2}= \frac{sin(\theta)cos(\theta)}{acos^2(\theta)+ bsin(\theta)cos(\theta)+ csin^2(\theta)}[/tex].

    Notice that there is no "r" in that! This can have a limit as r-> 0 only if it does NOT depend on [itex]\theta[/itex]- it is a constant. One obvious choice for a,b,c is a= c= 0, b= 1 but there may be others.
     
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